The Exact Analytical Solution And Conservation Law Of Several Kinds Of Partial Differential Equations

Many practical problems in the real world can be attributed to the partial differential equation model,and finding the solutions of the partial differential equation will help people to understand the practical problems better.Based on this,this thesis study several kinds of partial differential equation models based on Lie symmetry analysis method and supplemented by Maple.Firstly,the partial differential equation is reduced to ordinary differential equation by Lie symmetry analysis method;Then,according to the different types of equations,the corresponding solving methods are selected,such as power series method,extended hyperbolic tangent method,the simplest equation method,etc.,so as to obtain the exact analytic solution of the reduced equation;Finally,the conservation laws of the equation are established.The first chapter is the introduction part,which introduces the research background of partial differential equations,briefly introduces the current research status and research methods,points out the research significance of partial differential equations,and gives a brief overview of Lie symmetry analysis method and conservation law.The second chapter researches Benjamin-ono equation.Firstly,the five generators of the optimal system are deduced by Lie symmetry analysis in turn,ordinary differential equations that are easy to solve are obtained,and we use the power series method to calculate the exact analytical solutions of them.Finally,we establish conservation laws of the equation.In the third chapter,we study the coupled KdV-Burgers equation.Firstly,we obtain the Lie point symmetry based on the method of Lie symmetry analysis method.Then the power series theory is applied to solve the reduced equation and the power series solutions of the original equation are obtained;The solutions of the original equation in trigonometric form and hyperbolic form are obtained by using the(7)G?/G(8)-expansion method.In the end,the conservation laws of the equation are obtained by using the adjoint equation method.In the fourth chapter,the Bogoyavlensky-Konoplechenko equation is studied,which is a(2+1)dimensional equation.After two rounds of reduction of the equation with the method of Lie symmetry analysis,the ordinary differential equations which are easy to solve are obtained.In view of the characteristics of reduced ordinary differential equations,some exact analytical solutions are obtained by using the extended hyperbolic tangent method and the simplest equation method.In the fifth chapter,a six order generalized time fractional Sawada-Kotera equation is studied.Based on the fractional order calculus theory knowledge,we get some reduced ordinary differential equations by using Lie symmetry analysis method,and we give the proof process in detail.Then we use the power series method to obtain exact analytical solutions of the equation.Finally,according to the different order ?,and the conservation laws of equation are discussed.The sixth chapter is the summary and the prospect,we summary the research results,point out the unsolved problems,and look forward to the future development prospect of partial differential equations.